Regardless of the approach or approximation discussed above, the long-range nature of the quasiperiodic potentials does not allow us to apply Bloch's theorem to obtain quasimomenta as good quantum numbers and thus simplify the solution of the problem. Similarly, we cannot solve exactly the spectrum of the system on a computer as the quasiperiodicity manifests as a deviation from periodicity at all length scales, i.e., a true quasiperiodic system is infinitely large. Similar challenges also occur in the study of disordered systems. There the thermodynamic properties of the disordered material are approached by studying the impact of the disorder on a finite system and observing how the effect scales with the system's size. Crucially, a disorder-average is taken in the analysis, such that the general (noise) ensemble properties are explored, and not specific (sample path) cases. In comparison, for quasiperiodic systems, we can also study the spectral properties of finite-sized samples from the long-ranged system, and average over them, with a reduced statistical averaging arising from the self-similar nature of quasicrystalline eigenstates, see, e.g., Ref. [1]} and references therein.
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